KnowledgeBoat Logo
|

Mathematics

If 2=1.4 and 3\sqrt{2} = 1.4 \text{ and } \sqrt{3} = 1.7, find the value of each of the following, correct to one decimal place :

(i) 13−2\dfrac{1}{\sqrt{3} - \sqrt{2}}

(ii) 13+22\dfrac{1}{3 + 2\sqrt{2}}

(iii) 2−33\dfrac{2 - \sqrt{3}}{\sqrt{3}}

Rational Irrational Nos

21 Likes

Answer

(i) Given,

⇒13−2\Rightarrow \dfrac{1}{\sqrt{3} - \sqrt{2}}

Rationalizing,

⇒13−2×3+23+2⇒3+2(3)2−(2)2⇒3+23−2⇒3+21⇒3+2⇒1.7+1.4⇒3.1\Rightarrow \dfrac{1}{\sqrt{3} - \sqrt{2}} \times \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} \\[1em] \Rightarrow \dfrac{\sqrt{3} + \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} \\[1em] \Rightarrow \dfrac{\sqrt{3} + \sqrt{2}}{3 - 2} \\[1em] \Rightarrow \dfrac{\sqrt{3} + \sqrt{2}}{1} \\[1em] \Rightarrow \sqrt{3} + \sqrt{2} \\[1em] \Rightarrow 1.7 + 1.4 \\[1em] \Rightarrow 3.1

Hence, 13−2\dfrac{1}{\sqrt{3} - \sqrt{2}} = 3.1

(ii) Given,

⇒13+22\Rightarrow \dfrac{1}{3 + 2\sqrt{2}}

Rationalizing,

⇒13+22×3−223−22⇒3−2232−(22)2⇒3−229−8⇒3−221⇒3−22⇒3−2×1.4⇒3−2.8⇒0.2\Rightarrow \dfrac{1}{3 + 2\sqrt{2}} \times \dfrac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} \\[1em] \Rightarrow \dfrac{3 - 2\sqrt{2}}{3^2 - (2\sqrt{2})^2} \\[1em] \Rightarrow \dfrac{3 - 2\sqrt{2}}{9 - 8} \\[1em] \Rightarrow \dfrac{3 - 2\sqrt{2}}{1} \\[1em] \Rightarrow 3 - 2\sqrt{2} \\[1em] \Rightarrow 3 - 2 \times 1.4 \\[1em] \Rightarrow 3 - 2.8 \\[1em] \Rightarrow 0.2

Hence, 13+22\dfrac{1}{3 + 2\sqrt{2}} = 0.2

(iii) Given,

⇒2−33\Rightarrow \dfrac{2 - \sqrt{3}}{\sqrt{3}}

Rationalizing,

⇒2−33×33⇒23−33⇒2×1.7−33⇒3.4−33⇒0.43⇒0.1\Rightarrow \dfrac{2 - \sqrt{3}}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} \\[1em] \Rightarrow \dfrac{2\sqrt{3} - 3}{3} \\[1em] \Rightarrow \dfrac{2 \times 1.7 - 3}{3} \\[1em] \Rightarrow \dfrac{3.4 - 3}{3} \\[1em] \Rightarrow \dfrac{0.4}{3} \\[1em] \Rightarrow 0.1

Hence, 2−33\dfrac{2 - \sqrt{3}}{\sqrt{3}} = 0.1

Answered By

9 Likes

Related Questions